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Regarding Q17, half of 324,000

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Which number is greater: 2^{300} or 3^{200} ?

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The tuesday problem and when no analytical solution exists.

So first the tuesday problem. It goes like this:

The chance of getting a boy is 50%, the possible outcomes are either a boy or a girl. A person have two kids. One of them is a boy. One of them are born on a tuesday. What are the chances the second kid is a girl?

So, the answers should be 14/27 for the second being a girl if you know it's on a tuesday the boy is born and if you don't know what day of the week, then the chance is 2/3 for the second being a girl.

Now, I don't really agree with this. As I see it, the chance is either 1 or 0, because this person already have two kids, the event already happened. On the other hand, had the question been, if you were to guess what gender my other kid is based on information X, then I'd agree with the probabilities given.

I think people automatically thinks the situation is there's one unborn child and then needs to find the probability of the unborn child being a girl, but if one child was unborn, then the chance would be 50% girl/boy and the day of the week wouldn't matter, I think, because now it's not "one is a boy", no it's "the first one is a boy".

This makes all the difference!

Let me illustrate:
If you know nothing, except for two kids, the possibilities are:
GB, BG, BB, GG. [It means, first kid girl, second boy, etc.]
A) If you know the first one is a boy, the possibilities are:
BG, BB.
B) If you know one of the two is a boy, the possibilities are:
BB, GB, BG. [In reality, there's only one possibility, because the outcome has already happened].

So for case A) it's 50/50 and for case B) it's 2/3 for girl and 1/3 for boy, if you were to guess. [Again, it has already happened, case B) is merely finding out the best thing to guess for, not what is the most likely child to be born, because childs are born in a differentable order, kid 1, kid 2. In the case of case B) however you can't seperate kid 1 and kid 2 and therefore have to count both cases].

Now adding the weekday information. If we go back to case A), where you know the first is a boy, the weekday makes no difference, because the possible outcomes becomes:

Boy burn on tuesday followed by boy born some day contra girl born some day. The day, tuesday, does not change the possible outcomes of the second child.

However, in the case of the child already been determined and therefore you don't know if it's the first or second child you've information about, the information of the weekday changes the chances of guessing correct. It's because the weekday information limits the possible outcomes, like this:

Either it's BG:
Then the boy is born a tuesday and the girl in one of the seven weekdays (7 possibilities).
Or it's GB:
Then the boy is born a tuesday and the girl in one of the seven weekdays (7 possibilities).
Or it's BB, however before BB only occured in one way (whereas GB and BG occoured in twice as many ways, which is still true, if the boy could have been born on any given day of the week, which I'll show in a moment), now it occours in many ways:
One way is Boy born on tuesday, followed by boy born any weekday (7 possibilities), the other being boy born on any weekday, followed by boy born on a tuesday (now 6 possibilities, because tuesday/tuesday is already counted). In total this gives 27 possibilities and the chance of a girl is 7+7, which means the tuesday information changes the outcome of you guessing correct, if you guessing a girl to 14/27 in stead of 2/3.

So let's look at the boy being born on any given day of the week, as I said I would:
Boy born on any day, followed by girl born on any day. The possibilities are 7*7, 49.
Next, girl born on any day, followed by boy born on any day, again 49 possibilities.
Now for BB, there are boy born on any day, followed by boy born on any day, the possibilities are:
7*7=49 possibilities.

The difference lies in, that the tuesday information for case A) did not remove any possible outcomes, it did for case B), where there's no difference for the girl outcomes, for the boy outcome, the following possibilities can be listed:
MM
MT
MW
MT
MF
MS
MS
TM
TT
TW
TT
TF
TS
TS
WM
WT
WW
WT
WF
WS
WS
TM
TT
TW
TT
TF
TS
TS
FM
FT
FW
FT
FF
FS
FS
SM
ST
SW
ST
SF
SS
SS
SM
ST
SW
ST
SF
SS
SS

Now, for case A), that list goes down to 7 possibilities, because you know the first one is a boy [on a tusday, so all starting with a T (that is not thursday)], and so the possible days for a girl also goes to 7 possibilites. However for case B), the list changes from all those 49 possibilities, with no information about the weekday, to all that contains a tuesday, when the information about tuesday is added, and that is in total 13 possibilities that contains a tuesday:
MT
TM
TT
TW
TT
TF
TS
TS
WT
TT
FT
ST
ST

This does however seem counter intuitive. So you're to guess if the second kid is a girl or a boy and wants to find out what is the best guess, i.e. what guess gives you the right answer most times. You know that one of the kids is a boy, and you know that the boy is born on a given day of the week, it might as well be tuesday, it might be any day of the week though. So it does seem weird that getting told it's a tuesday should alter the information, would a proper way of dealing with the problem not be to assume the boy is born on a given weekday in the first place, realise that there's no difference what weekday is choosen [it always goes from 49 to 13 possibilities] and then choose one at random? For yes, the boy could be born on any given day of the week, but no matter what day you're told, the boy is born on one of those days, which means all other possibilities becomes impossible to the degree of assumptions put on the system and therefore, I'd think, should not be counted.

So to me, it seems like, the initial question, the one without information about the day "I have two kids, one is a boy, if you guess girl, what are the chances you guess correct", does not take into consideration that the boy already is born on a given day and is not going to be born on a certain day. Therefore, I'd say no matter the information of birth on a tuesday, the probability should be 13/27 for boy, 14/27 for girl. Heck maybe there are even more assumptions that should be added, more information, and maybe it actually is 50/50?

What I mean is that in the BG/GB/BB Scenario, it's assumed that BG,GB and BB are picked equally often, but looking through the weekdays they can be born and knowing the boy is born on a given weekday, it reduces it to a 13/27 contra 14/27 probability for guessing correct, not a 1/3 contra a 2/3.

Then again, if one takes into account that one knows the other child, boy or girl, is born on a given weekday as well, then for born on a tuesday it becomes 1/3 contra 2/3 and for all other days it becomes 50/50.

Hmm, I'm confused, hopefully someone can make it more clear.



Now to the next thing I have been thinking about. I have often been told there are equations of which there are no analytical solution. To which I wonder, is that not simply because we don't have a set of functions to describe any possible curve. I'd guess if we'd a set of functions that could describe any possible curve, then we could solve every equation analytical. I'd guess so because every solution to a given equation always is in the form of a given n-dimensional curve, to my knowledge. If we had a set of functions that could describe any curve, then the right combination of these functions would always be able to create such a curve, in return meaning that the solution is analytical, I'd think.
The type of functions I know about are, to my knowledge, not designed due to any geometrical advantages, these functions are those general from operations such as sum, product (which basicly just is a more generalized way of sum, I believe), and potens/exponent (which basicly is just a more generalized way of product, I believe). Then there are the sinus and cosinus functions which are closely related to the exponent functions through imaginable numbers, which in return means, if one expands the set of possible input numbers to imaginary, these functions are as well part of the exponent functions, a^x.

However, as it is, to my knowledge, no combination of any of these types of functions can describe a sharp change in a curve, like what I try to draw below:
     /
___/
This could be described as the function y = k [k is constant] from x = a to x = b, where a and b are some numbers.
And y = x for x = b to x = c, where b and c again are some numbers.
But the unified function from a to c can, to my knowledge, not be described through the means of any of those functions I listed. Though, I can imagine, a fourier transformation could maybe bring one very, very, close, since I specified a closed area, but if the functions were global (-infinity to b,y=k and b to infinity,y=x). I'd imagine a set of functions designed to describe any type of geometrical shape/curve in a n-dimensional space would on the other hand be able to solve this problem. Though then there's of course, when going from curve, to shape, the whole problem of several y-values for a single x-value (like the squareroot of any non-zero number provides two possible outcomes, Y and -Y). Though if at least every possible curve where described, then every equation would have an analytical solution, I'd guess. [Maybe it already has, just only approximative? As mentioned with the fourier transformation, one can expand a function in a potens base (taylor), or maybe a exponential base (fourier) and then solve for the function in the local region, but the approximation, not only have some assumptions needs to be meet, (like I believe the taylor expansion requires the function to be differentiable in every single point), but will always be approximative [as it gets closer and closer to the given region of space selected] and therefore, not really useable, if you want to know what happens values of x's far away (I'm not sure, is this where chaos theory originates from?). So all in all, I think it'd be pretty nice with such a set of functions that could describe any possible curve in an n-domensional space.

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